First-principles Calculations

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  • Sains Malaysiana 45(10)(2016): 15511556

    Pressure Dependence of Structural, Elastic and Electronic Properties of -Al2O3: First-principles Calculations

    (Tekanan Pergantungan kepada Sifat Struktur, Anjal dan Elektronik -Al2O3: Pengiraan Prinsip-Pertama)



    The first-principles calculations were performed to investigate the structural, elastic, mechanical and electronic properties of -Al2O3 at applied pressure up to 50 GPa. The obtained ground state properties were in agreement with previous experimental and theoretical data. The elastic constants, bulk modulus, shear modulus, Youngs modulus and anisotropy have been calculated as pressure increased. It was found that there was a brittle-ductile transition at about 23.2 GPa. The increasing ratio Ba /Bc with pressure indicates the weakening chemical bonding and the increasing anisotropy in this compound. The electronic structures were also calculated, which shows that band gaps increase monotonically. The population analysis showed the charge transfer was mainly between Al-3s and O-2p as pressure increased.

    Keywords: Density functional theory; elastic properties; electronic structure; -Al2O3


    Pengiraan prinsip-pertama dijalankan untuk mengkaji sifat struktur, anjal, mekanik dan elektronik -Al2O3 pada tekanan yang dikenakan sehingga 50 GPa. Sifat keadaan tanah yang diperoleh adalah sama dengan data uji kaji dan teori yang terdahulu. Pemalar anjal, modulus pukal, modulus ricih, modulus Young dan anisotrofi telah dihitung apabila tekanan meningkat. Didapati bahawa terdapat peralihan rapuh-mulur pada 23.2 GPa. Peningkatan nisbah Ba /Bc dengan tekanan menunjukkan ikatan kimia yang semakin lemah dan anisotrofi yang semakin meningkat dalam sebatian ini. Struktur elektronik juga dihitung yang menunjukkan bahawa jurang jalur meningkat secara senada. Analisis penduduk menunjukkan pemindahan caj antara Al-3s dan O-2p apabila tekanan meningkat.

    Kata kunci: Sifat anjal; struktur elektronik; teori fungsi ketumpatan; -Al2O3


    Al2O3 is an extremely important ceramic material to industrial applications (Chen et al. 2008; Kruse et al. 1996; Matsunaga et al. 2003; Zhukovskii et al. 2001). The many applications alumina are its use as a coating for cemented carbide cutting tools, high- gate dielectrics, multilayer structures with high damage thresholds in UV laser applications, heterogeneous catalysis, corrosion protection and thermal barriers coating (Ahuja et al. 2004; Copel et al. 2001; Fernndez et al. 2003, 2003b; Gusev et al. 2001; Haverty et al. 2002; Hosseini et al. 2005a, 2005b; Liao et al. 1999; Ollivier et al. 1997; Shang et al. 2007; Shi et al. 2006; Vali & Hosseini 2004; Yang et al. 2009, 2004) due to its high band gap and band offset properties, high dielectric constant, high hardness, high mechanical strength, good abrasion resistance, good corrosion resistance and good electrical insulation. Among those applications, Al2O3 as a thermal barrier coating has been paid much attention (Limarga et al. 2005, 2002; Shanmugavelayutham et al. 2006; Widjaja et al. 2002; Yu et al. 2010). This application is needed to measure and monitor the change of stress in the thermally grown oxide layer to understand the oxidation resistance, where the elastic properties that show the response to an applied

    macroscopic stress are required to study. On the other hand, the basic thermodynamic variable of pressure is used to transport matter and develop the micromechanical devices. Moreover, as a promising gate dielectric material in metal-oxide-semiconductor (MOS) devices, it is important to study the pressure dependence of the band gaps because the electronic properties of strained lattices can be obtained by investigating the effects of strain on bulk samples. Theoretically, there are many works to study Al2O3, e.g. the first-principles plane-wave pseudopotential calculations with VASP code (Matsunaga et al. 2003), the full potential-linearized augmented plane wave method with WIEN2k package (Hosseini et al. 2005b), the first-principles pseudopotentials and plane waves with ABINIT code (Vali & Hosseini 2004), the first-principles density functional calculations with SIESTA code (Fernndez et al. 2003b) and the full-potential linear muffin-tin-orbital (FPLMTO) method (Ahuja et al. 2004). However, the application of pressure on Al2O3 to study the pressure dependence of the structural, elastic and electronic properties was not taken into account. Therefore, the investigation Al2O3 under pressure is essential. In this paper, we show the first-principles calculations to investigate the properties of -Al2O3 under pressure.

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    The calculational methods are shown in the next section, the results and discussion are presented next and the conclusions are given in the last section.



    The calculations are performed using the plane-wave ultrasoft pseudopotential with the generalized gradient approximation (GGA) as parameterized by Perdew et al. (1992) with CASTEP code (Segall et al. 2002). The kinetic energy cutoff of 380 eV and the k-point meshes of 666 for -Al2O3 have been set to structure optimization and energy calculation. The Al 3s2, 3p1, O 2s2, 2p4 electrons are explicitly treated as valence electrons. The convergence criterion for the maximal force between atoms is 0.01 eV/, the maximum stress is 0.02 GPa and the maximum displacement is 5.0104.


    The elastic constants are important parameters that describe the response to an applied macroscopic stress, which are defined by means of a Taylor expansion of the total energy E(V, ) for a strained system of volume V with respect to a strain parameter (Chen 1996; Fast et al. 1995; Tian 2004). As a matter of fact, the structure of -Al2O3 has six independent elastic constants c11, c12, c13, c14, c33 and c44. The mechanical stability conditions for -Al2O3 under pressure are known as (Sinko & Smirnov 2002):


    where = c P(=1, 3, 4), 12 = c12 + P, 13 = c13 + P. The Voigt approximation has proposed the averaging of relations expressing the stress in terms of the given strain (Voigt 1928) and the Reuss approximation has proposed the averaging of the relations expressing the strain in terms of the given stress (Reuss 1929). Hill (1952) has proved that the Voigt and Reuss equations represent upper and lower bounds of elastic constants, respectively. Hence, he took an arithmetic mean value of the two approaches as follows:



    where sij is the inverse matrix of the elastic constants

    matrix cij.

    Then, the Youngs modulus E is calculated using (4):




    -Al2O3 is a trigonal crystal system with space group and local symmetry . In order to determine the

    equilibrium geometry, we first calculate the structural properties using the obtained total energy as a function of volume and the third-order Birch-Murnaghan equation of state (Murnaghan 1944). The calculated results are summarized in Table 1 together with previous available theoretical (Duan et al. 1999, 1998; Matsunaga et al. 2003; Shang et al. 2007; Ru & Qiu 2009) and experimental (dAmour et al. 1978; Lee & Lagerlof 1985) data. Obviously, we can see that our results are in agreement with the experiments and calculations. Figure 1 displays the pressure dependence of the normalized structural parameters a/a0, /0 and V/V0 from 0 to 50 GPa (a0, 0 and V0 are experimental data). As pressure increasing, the a/a0 and V/V0 decrease whereas /0 increases. The best fit of these curves show an almost linear behavior presented in Figure 1.

    FIGURE 1. The pressure dependence of the normalized structural parameters a/a0, /0 and V/V0 from 0 to 50 GPa


    Elastic constants are important parameters of materials, give worth information to study the structural stability, determine the response of the crystal to external forces and provide a link between the mechanical and dynamical behaviors of crystals. The calculated six independent elastic constants are listed in Table 1 together with the available experimental (Gieske & Barsch 1968; Gladden et al. 2004; Hovis et al. 2006) and theoretical (Duan et

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    al. 1999; Ru & Qiu 2009; Shang et al. 2007) data at 0 GPa and 0 K. According to the Born stability conditions at 0 GPa for trigonal structures (C44 > 0, C11 > C12, C33 (C11 + C12) > (Born & Huang 1982)), our calculated results satisfy these criteria, which indicates -Al2O3 was mechanically stable. We also calculate the bulk modulus, shear modulus, Youngs modulus, Youngs modulus along the x- and z-axis, B/G using (2)-(4), which are given in Table 1. We can see that our results are consistent with previous theoretical (Duan et al. 1999; Shang et al. 2007) and experimental (Gieske & Barsch 1968; Gladden et al. 2004; Hovis et al. 2006) data. The parameter limiting the stability of this compound is shear modulus due to B>G. As we know, the bulk modulus represents the resistance to fracture and the shear modulus represents the resistance to plastic deformation. A material is ductile (brittle) if the B/G is more (less) than 1.75 (Pugh 1954). In this case, this compound is described as brittle. The mechanical anisotropy is computed by using the bulk moduli Ba and Bc (shown in Ref. (Ravindran et al. 1998; Zhu et al. 2008)) along the a- and c-axis, respectively. The ratio Ba /Bc is 1.109 which indicates this compound has strong chemical bonding and anisotropy. Elastic properties under pressure are studied in detail. The pressure dependence of elastic constants is shown in Figure 2, which shows the elastic constants incr